The one-dimensional problem of bin packing (see [1]) can be formulated as follows. ... approximation algorithm and the expected optimal number of bins (see ...
Russian Mathematics (Iz. VUZ) Vol. 41, No. 12, pp.25{33, 1997
Izvestiya VUZ. Matematika UDC 519.852
BIN PACKING: ASYMPTOTICALLY EXACT APPROACH E.Kh. Gimadi and V.V. Zalyubovski
1. The one-dimensional problem of bin packing (see [1]) can be formulated as follows. Given a list of objects (items) S = {ai | i = 1, . . . , n} and the bin capacity B , it is required to pack the objects from the list S into minimal number of bins in such a manner that the sum of weights of the objects in each bin not exceed B . In what follows the notation ai stands for both an object with the number i and its weight. It is intrinsic to suppose that 0 < ai ≤ B for all i = 1, . . . , n. This problem is N P -dicult (see [2]). Already in rst works dealing with the study of bin packing, there was proposed a number of low-cost approximation algorithms and obtained guaranteed bounds for its relative error in the worst case (ibid.). There was noted also that the behavior of these algorithms \on the average" essentially di ers from the obtained bounds. Seemingly, this fact stimulated the appearance of works dealing with probabilistic investigation of the problem. As a rule, those works contain either an estimation of the ratio between the mean value of the number of bins required by certain approximation algorithm and the expected optimal number of bins (see [3]{[4]), or an investigation of the probabilistic properties of the optimal solution (see [5]). In the present article we investigate the quality of the approximation algorithm by means of the idea of constructing algorithms with (ε, δ )-estimations which was proposed in [6]. The essence of such an approach is presented in Section 2. In Section 3 we describe the distribution functions for weights of objects, which are considered in what follows. In this case we talk about so-called B -asymmetric and B -regular lists. In Section 4 we describe an approximation algorithm A for solving bin packing problem and give the proof of the optimality of this algorithm in the case of determined B -asymmetric and B -regular lists. In Sections 5{7 we investigate the behavior of a modi ed approximation algorithm Ae on lists with random weights of objects. In Section 5 we prove, for example, a theorem on the upper bound for the probability of the algorithm Ae failing. In Sections 6 and 7 we establish the conditions of asymptotical exactness of Ae as concerns the B -asymmetric and B -regular distribution functions, respectively.
2. Let
A be a certain algorithm for solving optimization problem on minimum. We denote by and F ∗ (S ) the meanings of the goal functions on the solution of a problem S obtained by means of the algorithm A and on the optimal solution, respectively. We say that the algorithm A satis es estimations (εn , δn ) on a class of problems Kn of the dimension n if for all S ∈ Kn there is valid the inequality © ª P r FA (S ) > (1 + εn )F ∗ (S ) ≤ δn .
FA (S )
The work was supported by Russian Foundation for Basic Research, grants 96-01-01591 and 97-01-00890. c 1997 by Allerton Press, Inc. °
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The algorithm A is said to be asymptotically exact on a class of problems K if there exist sequences (εn ), (δn ) such that for any n the algorithm A satis es the estimations (εn , δn ) on a subset Kn ⊂ K of problems of the dimension n and εn → 0, δn → 0 as n → ∞. One can consider the parameters (εn ) and (δn ) as bounds for the relative error and the probability of failing of the algorithm, respectively. The technique of (εn , δn )-estimations made it possible to establish conditions for asymptotical exactness of the low-cost approximation algorithms for solving a number of dicult problems of discrete optimization (see [7]{[12]), including the bin packing problem for random lists with the symmetric and non-increasing functions of weight distribution (see [9], [11]). In the present article we investigate a modi ed algorithm for bin packing and justify its asymptotical exactness for a class of problems which is essentially more extensive.
3. In what follows we suppose that ai ∈ IB , where IB = {1, 2, . . . , B} (B is integer). Let us introduce into consideration the characteristic function (c. f.) χS of a list S : χS (r) | the quantity of objects of the weight r in the list S , r ∈ IB . We shall use the notation: bxc is the most among integers which do not exceed x; dxe is the least among integers which are greater than or equal to x; {x} is the fractional part of x ({x} = x − bxc). A non-negative function f of integer-valued argument r ∈ Ib is said to be b-asymmetric (b{ symmetric) if f (r) ≤ f (b − r) (f (r) = f (b − r)) for r = 1, . . . , bb/2c and f (r) = 0 for r > b. Obviously, the class of B -asymmetric functions includes also the B -symmetric functions, the non-decreasing functions, the functions of a constant value. In addition, the asymmetry of a list is equivalent to its representability in the form of the maximal B -symmetric sublist and a remaining sublist of objects with their weights exceeding B/2. A function f is said to be B -regular for B = 2Q , Q is integer, if it can be represented as the sum of 2k -symmetric functions f (k) , k ∈ IQ , f (r) =
Q X k=1
f (k) (r),
r ∈ IB .
The decomposition of a B -regular function f into 2k -symmetric components f (k) , found via the following recurrent relations: f (k) (r)
k fk (2 − r)
= fk (r) 0
for 1 ≤ r < 2k−1 ; for 2k−1 ≤ r ≤ 2k ; for r > 2k ,
(
fk−1 (r) =
fk (r) − f (k) (r)
0
k ∈ IQ ,
can be
for 1 ≤ r ≤ 2k−1 ; for r > 2k−1 , (1)
where fQ (r) = f (r) for r ∈ IB . Relations (1) imply: rst, that the functions fk are the 2k -regular ones; second, the necessary and sucient condition for the B -regularity of a function f is the ful llment of the inequalities: fk (r) ≥ f (k) (2k − r), r = 1, . . . , 2k−1 , k ∈ IQ ; third, the decomposition of a B -regular function can be written in the form f (r) =
where
Q X k=kr
f (k) (r),
r ∈ IB ,
= Q; kr = min{ k | r < 2k } = dlog2 (r + 1)e, 1 ≤ r < B. An important subclass of the B -regular functions is the set of the non-increasing functions f (r), r ∈ IB , where B = 2Q , Q being an integer. kB
24
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Lemma 1 ([9]). A non-increasing function f (r),
B -regular.
r ∈ IB ,
(where
B
= 2Q ,
Q
is integer) is
4. By a b-bundle we shall call either one object with the weight b or a pair of objects with the weights r and b − r for r = 1, . . . , bb/2c. Let us describe an approximation algorithm A for bin packing of objects, which was suggested by the authors in [9]. If B = 2Q , Q is integer, then the algorithm A consecutively for k = Q, Q−1, . . . , 1 forms a list of 2k -bundles and packs this list into bins by the principle of the N F D-algorithm (\next feasible with preliminary ordering by lack of increase"). The objects from a sublist S 0 ⊂ S , which cannot be placed into full bins, are packed by means of the N F -algorithm (\next feasible"). The description of the algorithms N F D and N F can be found in [1], [2]. In the case of B -asymmetric lists and for B 6= 2Q , Q being integer, we use a simpli ed version of the algorithm A : we ll bins by B -bundles and locate the rest objects ai ∈ S 0 by means of the N F -algorithm. If B ≤ cn, where c > 0 is a constant, then the working time of the algorithm A is linear with P respect to n. It is clear that for {ai | i ∈ S 0 } ≤ B the algorithm A gives us the exact solution of the problem. Lemma 2 ([9]). For B -regular lists S , the algorithm A nds the exact solution of the bin packing problem with the value of the goal function FA = FS∗ being equal to »
1
B X
B r=1
¼
rχS (r) .
Lemma 3. For B -asymmetric lists S , the simpli ed algorithm A nds the exact solution of the
bin packing problem with the value of the goal function being equal to B X r =d(B +1)/2e
χS (r) +
»½
B+1
2
¾
¼
χS (bB/2c) .
Proof. Indeed, for B -asymmetric lists, the simpli ed algorithm A P packs the objects into a minimal number of bins. For odd B , this number is equal to the quantity {χS (r) | B/2 ≤ r ≤ B} of all those objects whose weight exceeds the half of the capacity of a bin. For even B , we must add to this number d0.5χS (B/2)e bins containing the objects of the weight B/2. Then we can write the additional term in the form d{0.5(B + 1)}χS (bB/2c)e, which gives us 0 and d0.5χS (B/2)e, for odd and even B , respectively.
5. Now we turn to a bin packing with random lists S = {ai | i ∈ In }, where the weights of objects are independent random variables with the discrete distribution function pr
= P r{ai = r} ≥ 0,
r
= 1, . . . , B ;
B X r =1
pr
= 1.
(2)
In what follows we shall consider the class Kn0 of bin packing problems with the following rule of forming the random lists: during n consecutive independent tests the next object ai gets into one of weight classes r = 1, . . . , B in accordance with the distribution function (2). In that situation the values χS (1), . . . , χS (B ) of the characteristic function of list S are, generally speaking, dependent P random variables because they are connected via relation {χS (r) | r ∈ IB } = n. There arises an intrinsic question on the possibility of low-cost algorithms for solving problems of the class Kn0 in the case where the distribution functions satisfy some properties similar to those which enable us to apply e ective exact algorithm A for the determined lists. This question 25
Russian Mathematics (Iz. VUZ)
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does not have a trivial answer, because, in the general situation, the corresponding property of the distribution function pr is not preserved for the characteristic functions χS (r) of a concrete realization of the random list. Let us note that, in the situation where the weights of objects are random variables, the de nition of the class of problems is related to the form of the distribution function (2). In order to solve the problems from the class Kn0 we apply a modi ed approximation algorithm Ae, which uses a solution obtained via the algorithm A for a certain estimative list Se under the condition that it majorizes the initial list S , i. e., for any weight class r = 1, . . . , B there are ful lled the inequalities χS (r) ≤ χSe(r).
(3)
In the set of random lists S we select a subset S of the lists satisfying with arbitrary r ∈ IB the following relations: |χS (r) − npr | ≤ r ,
(4)
where r = 2(npr (1 − pr ) ln n)1/2 . We apply the algorithm Ae only in the case where a concrete realization of a random list belongs to the set S . Otherwise we say that the algorithm fails. Thus, for a list from S , we apply rst the algorithm A to estimative list Se. Then we remove from full bins all objects which do not belong to the initial list S , and thus obtain solution of the problem with a number of bins which equals FAe. For estimation of the probability of the algorithm failing we need inequalities for probabilities n P of large deviations of sum Sn = ξi of the independent random variables ξ1 , . . . , ξn , which are i=1 distributed by the Bernoulli law P r{ξi = 1} = p; P r{ξi = 0} = 1 − p, i = 1, . . . , n. Lemma 4 ([13], Chap. 5, p. 131). For ≥ 0, P r{Sn − np ≥ } ≤ exp(−nH (p + /n)), P r{Sn − np ≤ −} ≤ exp(−nH (p − /n)), where H (x) = x ln(x/p) + (1 − x) ln((1 − x)/(1 − p)). By virtue of the relations 2 2 −nH (p + /n) = − + o(1), −nH (p − /n) = − + o(1), 2np(1 − p) 2np(1 − p) which are valid for = o(n2/3 ) (see [13], Chap. 5, p. 131), we obtain Corollary. For = o(n2/3 ), there is valid the inequality ³ 2 ´ P r{|Sn − np| ≥ } ≤ c exp − , 2np(1 − p) where c = 2(1 + o(1)).
Theorem 1. The probability of the algorithm Ae failing on the class of problems Kn0 is bounded
above by the value O(B/n2 ).
Proof. If Ar stands for the event de ned by (4), then for the probability of the algorithm Ae
failing we have
δAe = P r
B n[
where Ar means the event opposite to Ar .
r =1
o
Ar ≤
26
B X r =1
P r(Ar ),
Russian Mathematics (Iz. VUZ)
Vol. 41, No. 12
Since r ≤ 2(npr (1 − pr ) ln n)1/2 = o(n2/3 ), in accordance with Corollary of Lemma 4 we obtain that µ ¶ µ ¶ B B X X 2r 4np (1 − pr ) ln n cB δAe ≤ c exp − = c exp − r = 2. ¤ 2npr (1 − pr ) 2npr (1 − pr ) n r =1 r =1 Theorem 1 immediately implies the following
Corollary. If B = o(n/ ln n), then δAe = o
¡ 1 ¢ . n ln n
Now let us turn to the estimation of exactness of the algorithm Ae on the class of problems Kn0 .
6. In the case of a B -asymmetric function pr , r = 1, . . . , B , we apply the algorithm assumption that the characteristic function of estimative list Se is equal to χSe(r) = bnpr + r c,
Property (3) of the estimative list algorithm Ae functioning.
Theorem 2. The algorithm
function pr
Se
r
= 1, . . . , B.
Ae
under (5)
follows from de nition (5) and conditions (4) for the
on the class Kn0 of bin packing problems with has the following bound of relative error Ae
εAe = O
µs
B n/ ln n
B -asymmetric
¶
and is asymptotically exact for B = o(n/ ln n).
Proof. Note that, in the case r = 2(npr (1 − pr ) ln n)1/2 , the B -asymmetry of the function pr
yields the same property for the vector r , because in that situation (pr − pB−r )(1 − pr − pB−r ) ≤ 0, or pr (1 − pr ) ≤ pB−r (1 − pB−r ), and, consequently, r ≤ B−r , r = 1, . . . , bB/2c. Therefore, the characteristic function (5) of the estimative list Se is B -asymmetric as the integer part of a linear combination of the two B -asymmetric functions. Let us estimate the relative error εAe = (FAe − FS∗ )/FS∗ of the algorithm Ae. For convenience, we introduce the notation r xr =
B X r =d(B +1)/2e
xr .
As a lower bound for the minimal value of the goal function for odd B we take the number r χS (r) (i. e., the quantity of objects which have weights greater than the half of bin capacity). For even B we add the value χS (B/2)/2 to this number. Due to (4) this gives us the following lower bound FS∗ ≥ r χS (r) + χS
µ¹
B
º¶½
2
B+1
2
≥ r (npr − r ) + (npbB/2c − bB/2c ) µ
= n r pr + pbB/2c
½
27
B+1
2
¾¶
½
¾
≥
B+1
2
− r r .
¾
=
Russian Mathematics (Iz. VUZ)
Vol. 41, No. 12
Now let us obtain the upper bound for the value of the goal function after applying the algorithm Ae: » µ¹ º¶½ ¾¼ B B+1 FAe ≤ FSe∗ = r χSe(r) + χSe ≤ 2 2 ½ ¾ B+1 1 ≤ r (npr + r ) + (npbB/2c + bB/2c ) + . 2 2 We estimate εAe with regard for the fact that B -asymmetric function satis es the inequality r pr + pbB/2c {(B + 1)/2} ≥ 1/2. As a result, we have r (npr + r ) + (npbB/2c + bB/2c ){ B2+1 } + 12 2r r + 1/2 © B +1 ª . εAe ≤ −1≤ n/2 − r r n(r pr + pbB/2c 2 ) − r r √
√ npr ln n
and r √pr ≤ B , we obtain that s √ µs ¶ 1 + 8√n1ln n 4 n ln n r √pr + 1/2 B B √ =O . √ ≤ 8 n/ ln n ¡ ¢ n/ ln n n/2 − 2 n ln n r pr 1 − 4 n/Bln n 1/2
Owing to the inequalities r ≤ 2 εAe ≤
By virtue of Corollary of Theorem 1 we have δAe = o(1/(n ln n)). Thus, the algorithm Ae under assumptions of this proposition satis es the bounds εAe → 0, δAe → 0 as n → ∞. Consequently, it is asymptotically exact. Remark. If the maximal size of an object is bounded by the value R which is less than B , then we can re ne Theorem 2 by replacing B with R.
7. Now let us pass to the analysis of exactness of the algorithm Ae on the class Kn0 of problems with B -regular distribution function pr . We write out the decomposition of the function pr into 2k -symmetric components p(rk) , k = 1, . . . , Q, pr
=
Q X k=kr
p(rk) ,
r
= 1, . . . , B.
(6)
We use the following notation for the functions ϕ(x) = nx + 2(nx ln n)1/2 , 0 ≤ x ≤ 1; (
bϕ(p(rQ) )c dϕ(p(rk) )e
for k = Q, for 1 ≤ k < Q and de ne the list Se via its characteristic function gr(k)
=
χSe(r) =
Q X k=kr
gr(k) ,
r
(r = 1, . . . , B )
= 1, . . . , B.
(7)
Lemma 5. The list Se, de ned by (7), is B -regular one. If the algorithm Ae operates, then the
latter list is also an estimative list.
q
implies that the functions p(rk) are also 2k symmetric. Obviously, the functions ϕ p(rk) are 2k -symmetric as linear combinations of the 2k q symmetric functions p(rk) and p(rk) . Hence the 2k -symmetry of the functions gr(k) also follows. On the whole, these facts imply the B -regularity of the mentioned list Se by virtue of the representability of its characteristic function χS (r) in the form of the sum of 2k -symmetric functions.
Proof. The 2 -symmetry of the functions ¢ ¡ k
p(k) r
28
Russian Mathematics (Iz. VUZ)
Vol. 41, No. 12
Let us show that the list Se is the estimative one if the algorithm Ae operates. Indeed, in that case there are ful lled the inequalities χS (r) ≤ bnpr + r c, r = 1, . . . , B. On the other hand, formula (6) for decomposition of the distribution function and the inequalities ³ P ´ 1 /2 P 1 /2 xi ≤ xi , xi ≥ 0, justify the validity of inequalities (3): i
i
bnpr + r c = bnpr + 2(npr (1 − pr ) ln n)1/2 c ≤ j
≤ bnpr + 2(npr ln n)1/2 c = n j
≤ n
Q X
√
k=kr
p(rk) + 2 n ln n
≤
Q X k=kr
gr(k)
Q X k=kr
³
Q X
p(rk) + 2 n ln n
Q q k X (k)
=
pr
k=kr
= χSe(r),
j
k=kr
k=kr
= 1, . . . , B.
r
´ 1 /2 k
≤
k
Q X
n
p(rk)
ϕ(p(rk) ) ≤ ¤
We introduce the following notation ε~n
µ
=
B/αp n/ ln n
¶ 1 /2
,
B P
where αp = B1 rpr is the ratio of the mean value of the weight of objects and the capacity of bin r =1 which restricts the maximal possible weight of an object. Obviously, 0 < αp ≤ 1. Lemma 6. If a problem from the class Kn0 has B -regular distribution function and its list belongs to the set S , then it satis es the following lower bound for the optimal value of the goal function √ εn ). FS∗ ≥ nαp (1 − 3~
Proof. FS∗ ≥ ≥ nαp − ≥ nαp −
2√
2√
n ln n
B X
B r=1
n ln n
B µ
1
rχS (r) ≥
1
B X
B r=1
B X √ r =1
B (B + 1)
r pr ≥ nαp −
2√
n ln n
B
¶1/2
Bαp
r(npr − r ) ≥
≥ nαp −
µX B B X r =1
r
r =1
q
√
3n ln n
Bαp
¶1/2
rpr
≥ √
= nαp (1 − 3~εn ).
¤
2 Lemma 7. For any problem from the class Kn0 with B -regular distribution function, the approximation algorithm Ae with the estimative list Se (7) guarantees (in the case of its functioning) the upper bound for an approximate value of the goal function FAe ≤ nαp (1 + 2~ εn + ε~2n / ln n). B
Proof.
FAe ≤ ≤
1
FSe∗
µ B X
B r=1
≤
1
B X
B r=1
r Q − kr +
rχSe(r) + 1 ≤ Q X k=kr
¶
ϕ(p(rk) )
29
1
B X
B r=1
+1=
Q X
r
k=kr
1
B X
B r=1
gr(k) + 1 ≤ r (Q − kr ) +
Russian Mathematics (Iz. VUZ)
+
1
B X
B r=1
r
Q X k=kr
Vol. 41, No. 12 q
(np(rk) + 2
(k )
n ln npr
)+1=
1 B
(1 + 2 + 3 ) + 1,
where 1 = ≤
B X r =1
r(Q − kr ) =
Q− X1
Q− X1 k=1
(Q − k)
k 2X −1
r =2k−1
3 3 k (Q − k)4k ≤ B 2 8 k=1 8 k=1 4k
2 = n
B X r =1
r
∞ X
Q X k=kr
p(rk)
=n
B X r =1
r
r · pr
Q− X1
=
3 8
≤ B2
k=1
(Q − k)
3 · 2k−1 − 1 k−1 2 ≤ 2
∞ 0.5 · 2k 3 2 X 1 = B k 4 16 k=1 2k k=1 ∞ X
1
q
FAe ≤ nαp + 2 n ln nBαp + B/4 + 1 < µ
< nαp
s
1+2
B/αp n/ ln n
+
B/αp n
¶
= nαp (1 + 2~εn + ε~2n / ln n).
¤
Theorem 3. The algorithm Ae on the class Kn0 of bin packing problems with B -regular function
of distribution of weights is asymptotically exact for B/αp = o(n/ ln n).
Proof. Since B
≤ B/αp ,
we have that B = o(n/ ln n) and, in accordance with Corollary of Theorem 1, the probability of the algorithm Ae failing is bounded by the value o(1/(n ln n)) → 0 as n → ∞. Thus, our realization of the random list S satis es inequalities (3) and (4) with a probability tending to 1 as n increases. Let us estimate the error of the obtained solution with regard for Lemma 6. F − F∗ nαp (1 + 2~ εn + ε~2n / ln n) √ εAe = Ae ∗ S ≤ −1= FS nαp (1 − 3~ εn ) √ √ (2 + 3)~εn (1 + (2 − 3)~εn / ln n) √ = = O(~εn ) → 0 for n → ∞. ¤ 1 − 3~εn
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Russian Mathematics (Iz. VUZ)
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4. W. Knodel, A bin packing algorithm with the complexity O(n log n) and performance 1 in the stochastic limit, Lect. Notes Comput. Sci., Vol. 118, pp. 369{377, 1981. 5. W.T. Rhee and M. Talagrand, Optimal bin packing with items of random sizes. III, SIAM J. Comput., Vol. 18, no. 3, pp. 473-486, 1989. 6. E.Kh. Gimadi, N.I. Glebov and V.A. Perepelitsa, Algorithms with estimations for problems of discrete optimization, Problemy kibernetiki, Moscow, Vol. 31, pp. 35{42, 1975. 7. V.A. Perepelitsa and E.Kh. Gimadi, On the problem of determination of a minimal Hamilton way on a graph with weighted arcs, Diskretn. analiz, Novosibirsk, Vol. 15, pp. 57{65, 1969. 8. E.Kh. Gimadi and V.A. Perepelitsa, Asymptotically exact approach to solving traveling salesman problem, Upravlyaemye sistemy, Novosibirsk, Vol. 12, pp. 35{45, 1974. 9. E.Kh. Gimadi and V.V. Zalyubovski, Asymptotically exact approach to solving onedimensional bin packing problem, Upravlyaemye sistemy, Novosibirsk, Vol. 25, pp. 48{57, 1984. 10. E.Kh. Gimadi, Proof of a priori bounds for the quality of an approximate solving standardization problems, Upravlyaemye sistemy, Novosibirsk, Vol. 27, pp. 12{27, 1987. 11. , On certain mathematical models and methods for signi cant projects planning, Trudy Inst. Matem. SO AN SSSR, Novosibirsk, Vol. 10, pp. 89{115, 1988. 12. E.Kh. Gimadi, N.I. Glebov and A.I. Serdyukov, An algorithm for the approximate solving traveling salesman problem and its probabilistic analysis, Sibirsk. zhurnal issledovaniya operatsi, Novosibirsk, Vol. 1, no. 2, pp. 8{17, 1994. 13. A.A. Borovkov, Probability Theory, Nauka, Moscow, 1986.
03 February 1997
Institute of Mathematics of Siberian Branch of Russian Academy of Sciences
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